How to prove that the angle between the diagonal of cube and the diagonal of plane is 90 degrees?

Let the side length a be the root number 2*a and the diagonal number 3 * a.

If the face diagonal moves a/2 to the middle along the edge direction, it must intersect with the body diagonal, and then connect one end point of the body diagonal, and set the end point of the moved face diagonal as L to form a triangle with half of the face diagonal and half of the body diagonal.

Then: l 2 = a 2+(a/2) 2 = 5/4 * a 2.

The square of half the diagonal of a plane = (root number 2/2) 2 = 1/2 * a 2.

The square of the diagonal half of the object = (root number 3/2) 2 = 3/4 * a 2

In this triangle, the square of L is equal to the sum of the squares of half of the two diagonals, so it is a right triangle.

Therefore, the angle between the diagonal of the cube and the diagonal of the plane is 90 degrees.